#385416

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(5,6,7,8)(9,10)(11,19)(12,24)(13,27)(14,17)(15,22)(16,25)(18,28)(20,29)(21,31)(23,30)(26,38)(32,39)(33,34)(35,37)(36,40);(1,2)(3,4)(9,11)(10,15)(12,20)(13,21)(14,23)(16,29)(17,31)(18,30)(19,27)(22,28)(24,37)(25,34)(26,36)(32,40)(33,38)(35,39);(2,4)(9,12,13,25,14,26,28,39)(10,16,18,24,17,32,27,38)(11,20,21,34,23,36,22,35)(15,29,30,37,31,40,19,33);(10,17)(11,21,23,22)(15,19,31,30)(16,32)(18,27)(20,34,36,35)(24,38)(29,33,40,37)

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=efi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292612);